2.1 Complex numbers

Throughout this paper, we abbreviate $\otimes _{\mathbb {C}}$ as $\otimes $ and all varieties will be algebraic, complex, quasi-projective and reduced. If ${\mathcal {X}}$ is an irreducible variety, we denote by ${\mathcal {X}}^{\mathrm {nor}}$ its normalization. If ${\mathcal {X}}$ is an affine variety, we denote by ${\mathbb {C}}[{\mathcal {X}}]$ its coordinate ring: if, moreover, ${\mathcal {X}}$ is irreducible, then ${\mathcal {X}}^{\mathrm {nor}}$ is also affine and ${\mathbb {C}}[{\mathcal {X}}^{\mathrm {nor}}]$ is the integral closure of ${\mathbb {C}}[{\mathcal {X}}]$ in its fraction field (which is denoted by ${\mathbb {C}}({\mathcal {X}})$ ).

We fix in this paper a complex vector space V of finite dimension n. If X is a subset of V (or $V^*$ ), and if G is a subgroup of ${\mathbf G}{\mathbf L}_{\mathbb {C}}(V)$ , we denote by $G_X^{\mathrm {set}}$ (respectively, $G_X^{\mathrm {pt}}$ ) the setwise (respectively, pointwise) stabilizer of X and we set $G[X]=G_X^{\mathrm {set}}/G_X^{\mathrm {pt}}$ . Then $G[X]$ acts faithfully on X (and on the vector space spanned by X). If $X=\{v\}$ is a singleton, then $G_X^{\mathrm {set}}=G_X^{\mathrm {pt}}$ (and we denote both simply by $G_X$ or $G_v$ ) and $G[X]=1$ . If H is a subgroup of G, we set ${\overline {\mathrm {N}}}_G(H)={\mathrm {N}}_G(H)/H$ .

If, moreover, G is finite, we identify $(V^G)^*$ and $(V^*)^G$ , and we denote by $V_G$ the unique G-stable subspace of V such that $V=V_G \oplus V^G$ .

2.2 Reflections

Let W be a finite subgroup of ${\mathrm {GL}}_{\mathbb {C}}(V)$ . We set

$$ \begin{align*}{\mathrm{Ref}}(W)=\{s \in W~|\dim_{\mathbb{C}} V^s=n-1\}\end{align*} $$

and note that, for the moment, we do not assume that W is generated by ${\mathrm {Ref}}(W)$ . We set $\varepsilon : W \rightarrow {\mathbb {C}}^\times $ , $w \mapsto \det (w)$ . We identify ${\mathbb {C}}[V]$ (respectively, ${\mathbb {C}}[V^*]$ ) with the symmetric algebra ${\mathrm {S}}(V^*)$ (respectively, ${\mathrm {S}}(V)$ ).

We denote by ${\mathcal {A}}$ the set of reflecting hyperplanes of W, namely

$$ \begin{align*}{\mathcal{A}}=\{V^s~|~s \in {\mathrm{Ref}}(W)\}.\end{align*} $$

If $H \in {\mathcal {A}}$ , we denote by $\alpha _H$ an element of $V^*$ such that $H=\operatorname {{{\mathrm {Ker}}}}(\alpha _H)$ and by $\alpha _H^\vee $ an element of V such that $V=H \oplus {\mathbb {C}} \alpha _H^\vee $ and the line ${\mathbb {C}}\alpha _H^\vee $ is $W_H^{\mathrm {pt}}$ -stable. We set $e_H=|W_H^{\mathrm {pt}}|$ . Note that $W_H^{\mathrm {pt}}$ is cyclic of order $e_H$ and that $\operatorname {{{\mathrm {Irr}}}}(W_H^{\mathrm {pt}})=\{\operatorname {{{\mathrm {Res}}}}_{W_H^{\mathrm {pt}}}^W \varepsilon ^j~|~0 \hspace {0.1em}\mathop {\leqslant }\nolimits \hspace {0.1em} j \hspace {0.1em}\mathop {\leqslant }\nolimits \hspace {0.1em} e-1\}$ . We denote by $\varepsilon _{H,j}$ the (central) primitive idempotent of ${\mathbb {C}} W_H^{\mathrm {pt}}$ associated with the character $\operatorname {{{\mathrm {Res}}}}_{W_H^{\mathrm {pt}}}^W \varepsilon ^{-j}$ , namely

$$ \begin{align*}\varepsilon_{H,j}=\frac{1}{e_H}\sum_{w \in W_H^{\mathrm{pt}}} \varepsilon(w)^{\,j} w \in {\mathbb{C}} W_H^{\mathrm{pt}}.\end{align*} $$

If $\Omega $ is a W-orbit of reflecting hyperplanes, we write $e_\Omega $ for the common value of all the $e_H$ , where $H \in \Omega $ . We denote by $\nabla $ the set of pairs $(\Omega ,j)$ where $\Omega \in {\mathcal {A}}/W$ and $0 \hspace {0.1em}\mathop {\leqslant }\nolimits \hspace {0.1em} j \hspace {0.1em}\mathop {\leqslant }\nolimits \hspace {0.1em} e_\Omega -1$ . The vector space of families of complex numbers indexed by $\nabla $ is denoted by ${\mathbb {C}}^\nabla $ : elements of ${\mathbb {C}}^\nabla $ are called parameters. If $k=(k_{\Omega ,j})_{(\Omega ,j) \in \nabla } \in {\mathbb {C}}^\nabla $ , we define $k_{H,j}$ , for all $H \in \Omega $ and $j \in {\mathbb {Z}}$ , by $k_{H,j}=k_{\Omega ,j_0}$ where $\Omega $ is the W-orbit of H and $j_0$ is the unique element of $\{0,1,\ldots ,e_H-1\}$ such that $j \equiv j_0\, \mod e_H$ .

2.3 Parabolic subgroups

We denote by ${\mathrm {Parab}}(W)$ the set of parabolic subgroups of W (that is, the set of subgroups of W that are stabilizers of some point of V) and by ${\mathrm {Parab}}(W)/W$ the set of conjugacy classes of parabolic subgroups of W. If $P \in {\mathrm {Parab}}(W)$ , we denote by ${\mathcal {V}}(P)$ the set of elements $v \in V$ such that $W_v=P$ : it is a nonempty open subset of $V^P$ . By definition, $W_{V^P}^{\mathrm {pt}}=P$ and $W_{V^P}^{\mathrm {set}}={\mathrm {N}}_W(P)$ , so that $W[V^P]={\overline {\mathrm {N}}}_W(P)$ . The family $({\mathcal {V}}(P))_{P \in {\mathrm {Parab}}(W)}$ is a stratification of V (the order between strata corresponds to the reverse order of the inclusion of parabolic subgroups).

This stratification is stable under the action of the group W. If ${\mathfrak P} \in {\mathrm {Parab}}(W)/W$ , we denote by ${\mathcal {U}}({\mathfrak P})$ the image of ${\mathcal {V}}(P)$ in $V/W$ , where P is any element of ${\mathfrak P}$ . Then $({\mathcal {U}}({\mathfrak P}))_{{\mathfrak P} \in {\mathrm {Parab}}(W)/W}$ is a stratification of $V/W$ (the order between strata corresponds to the reverse order of the inclusion, up to conjugacy, of parabolic subgroups). Replacing V by $V^*$ , we similarly define ${\mathcal {V}}^*(P)$ and ${\mathcal {U}}^*({\mathfrak P})$ for $P \in {\mathrm {Parab}}(W)$ and ${\mathfrak P} \in {\mathrm {Parab}}(W)/W$ . By definition, ${\overline {\mathrm {N}}}_W(P)$ acts freely on ${\mathcal {V}}(P)$ or ${\mathcal {V}}^*(P)$ . Moreover, for $P \in {\mathfrak P}$ , the natural map ${\mathcal {V}}(P) \rightarrow {\mathcal {U}}({\mathfrak P})$ induces an isomorphism of varieties

$$ \begin{align*} {\mathcal{V}}(P)/{\overline{\mathrm{N}}}_W(P) \stackrel{\sim}{\longrightarrow} {\mathcal{U}}({\mathfrak P}). \end{align*} $$

In particular, ${\mathcal {U}}({\mathfrak P})$ is smooth.

2.4 Rational Cherednik algebra at ${\boldsymbol {\mathit {t}=0}}$

Let $k \in {\mathbb {C}}^\nabla $ . We define the rational Cherednik algebra ${\mathbf H}_k$ (at $t=0$ ) to be the quotient of the algebra ${\mathrm {T}}(V\oplus V^*)\rtimes W$ (the semi-direct product of the tensor algebra ${\mathrm {T}}(V \oplus V^*)$ with the group W) by the relations

(2-1) $$ \begin{align} \begin{cases} [x,x']=[\,y,y']=0,\\ [\,y,x]=\displaystyle{\sum_{H\in\mathcal{A}} \sum_{j=0}^{e_H-1} e_H(k_{H,j}-k_{H,j+1}) \frac{\langle y,\alpha_H \rangle \cdot \langle \alpha_H^{\scriptscriptstyle{\vee}},x\rangle}{\langle \alpha_H^{\scriptscriptstyle{\vee}},\alpha_H\rangle} \varepsilon_{H,j}}, \end{cases} \end{align} $$

for all x, $x'\in V^*$ , y, $y'\in V$ . Here $\langle \ ,\ \rangle : V\times V^*\rightarrow {\mathbb {C}}$ is the standard pairing. The first commutation relations imply that we have morphisms of algebras ${\mathbb {C}}[V] \rightarrow {\mathbf H}_k$ and ${\mathbb {C}}[V^*] \rightarrow {\mathbf H}_k$ . Recall [Reference Etingof and GinzburgEtGi, Theorem 1.3] that we have an isomorphism of ${\mathbb {C}}$ -vector spaces

(2-2) $$ \begin{align} {\mathbb{C}}[V] \otimes {\mathbb{C}} W \otimes {\mathbb{C}}[V^*] \stackrel{\sim}{\longrightarrow} {\mathbf H}_k \end{align} $$

induced by multiplication. (This the so-called Poincaré–Birkhoff–Witt decomposition, or PBW decomposition.)

2.5 Calogero–Moser space

We denote by ${\mathbf Z}_k$ the centre of the algebra ${\mathbf H}_k$ : it is well known [Reference Etingof and GinzburgEtGi, Theorem 3.3 and Lemma 3.5] that ${\mathbf Z}_k$ is an integral domain, which is integrally closed. Moreover, it contains ${\mathbb {C}}[V]^W$ and ${\mathbb {C}}[V^*]^W$ as subalgebras [Reference GordonGor, Proposition 3.6] (so it contains ${\mathbf P}={\mathbb {C}}[V]^W \otimes {\mathbb {C}}[V^*]^W$ ). If $W=\langle {\mathrm {Ref}}(W) \rangle $ , then ${\mathbf Z}_k$ is a free ${\mathbf P}$ -module of rank $|W|$ (see [Reference Etingof and GinzburgEtGi, Proposition 4.15]). We denote by ${\mathcal {Z}}_k$ the affine algebraic variety whose ring of regular functions ${\mathbb {C}}[{\mathcal {Z}}_k]$ is ${\mathbf Z}_k$ : this is the Calogero–Moser space associated with the datum $(V,W,k)$ . It is irreducible and normal.

We set ${\mathcal {P}}=V/W \times V^*/W$ , so that ${\mathbb {C}}[{\mathcal {P}}]={\mathbf P}$ and the inclusion ${\mathbf P} \hookrightarrow {\mathbf Z}_k$ induces a morphism of varieties

$$ \begin{align*}\Upsilon_k : {\mathcal{Z}}_k \longrightarrow {\mathcal{P}}\end{align*} $$

which is finite (and flat if $W=\langle {\mathrm {Ref}}(W) \rangle $ ).

2.6 Other structures on the Calogero–Moser space

The Calogero–Moser space ${\mathcal {Z}}_k$ is endowed with other structures (a ${\mathbb {C}}^\times $ -action, a Poisson bracket, a filtration, an action of ${\mathrm {N}}_{{\mathbf G}{\mathbf L}_{\mathbb {C}}(V)}(W)\ldots $ ) which are described below.

2.6.1 Grading, ${\mathbb {C}}^\times $ -action

The algebra ${\mathrm {T}}(V\oplus V^*)\rtimes W$ can be ${\mathbb {Z}}$ -graded in such a way that the generators have the following degrees:

$$ \begin{align*} \begin{cases} \deg(y)=-1 & \text{if }y \in V,\\ \deg(x)=1 & \text{if }x \in V^*,\\ \deg(w)=0 & \text{if }w \in W. \end{cases} \end{align*} $$

This descends to a ${\mathbb {Z}}$ -grading on ${\mathbf H}_k$ , because the defining relations (2-1) are homogeneous. Since the centre of a graded algebra is always graded, the subalgebra ${\mathbf Z}_k$ is also ${\mathbb {Z}}$ -graded. So the Calogero–Moser space ${\mathcal {Z}}_k$ inherits a regular ${\mathbb {C}}^\times $ -action. Note also that by definition ${\mathbf P}={\mathbb {C}}[V]^W \otimes {\mathbb {C}}[V^*]^W$ is clearly a graded subalgebra of ${\mathbf Z}_k$ .

2.6.2 Poisson structure

Let $t \in {\mathbb {C}}$ . One can define a deformation ${\mathbf H}_{t,k}$ of ${\mathbf H}_k$ as follows: ${\mathbf H}_{t,k}$ is the quotient of the algebra ${\mathrm {T}}(V\oplus V^*)\rtimes W$ by the relations

$$ \begin{align*} \begin{cases} [x,x']=[\,y,y']=0,\\ [\,y,x]=t \langle y,x \rangle + \displaystyle{\sum_{H\in\mathcal{A}} \sum_{=0}^{e_H-1} e_H(k_{H,i}-k_{H,i+1}) \frac{\langle y,\alpha_H \rangle \cdot \langle \alpha_H^{\scriptscriptstyle{\vee}},x\rangle}{\langle \alpha_H^{\scriptscriptstyle{\vee}},\alpha_H\rangle} \varepsilon_{H,i}}, \end{cases} \end{align*} $$

for all x, $x'\in V^*$ , y, $y'\in V$ . It is well known [Reference Etingof and GinzburgEtGi] that the PBW decomposition still holds so that the family $({\mathbf H}_{t,k})_{t \in {\mathbb {C}}}$ is a flat deformation of ${\mathbf H}_k={\mathbf H}_{0,k}$ . This allows us to define a Poisson bracket $\{\ ,\ \}$ on ${\mathbf Z}_k$ as follows: if $z_1$ , $z_2 \in {\mathbf Z}_k$ , we denote by $z_1^{t}$ , $z_2^t$ the corresponding element of ${\mathbf H}_{t,k}$ through the PBW decomposition and we define

$$ \begin{align*}\{z_1,z_2\} = \lim_{t \rightarrow 0} \frac{[z_1^t,z_2^t]}{t}.\end{align*} $$

Finally, note the following observation.

$$ \begin{align*} \textit{The Poisson bracket is }{\mathbb{C}}^\times\textit{-equivariant.} \end{align*} $$

2.6.3 Filtration

The tensor algebra ${\mathrm {T}}(V \oplus V^*)$ is naturally filtered by the subspaces $(\bigoplus _{j=0}^d (V \oplus V^*)^{\otimes j})$ . This induces a filtration of ${\mathrm {T}}(V \oplus V^*) \rtimes W$ by putting W in degree $0$ and so induces a filtration $({\mathcal {F}}_j {\mathbf H}_k)_{j \geqslant 0}$ of the rational Cherednik algebra. By convention, we set ${\mathcal {F}}_{-1} {\mathbf H}_k=0$ . If M is any subspace of ${\mathbf H}_k$ , we set ${\mathcal {F}}_j M=M \cap {\mathcal {F}}_j{\mathbf H}_k$ , so that M also inherits a filtration, and we denote by ${\mathrm {Rees}}_{\mathcal {F}} M$ the Rees module of M (associated with the filtration $({\mathcal {F}}_j X)_{j \geqslant 0}$ ), namely the ${\mathbb {C}}[\hbar ]$ -submodule of ${\mathbb {C}}[\hbar ] \otimes M$ equal to

$$ \begin{align*}{\mathrm{Rees}}_{\mathcal{F}} M=\bigoplus_{j \hspace{0.1em}\mathop{\geqslant}\nolimits\hspace{0.1em} 0} \hbar^{\kern1.2pt j} {\mathcal{F}}_j M.\end{align*} $$

Recall that, if $\lambda \in {\mathbb {C}}$ , then

(2-3) $$ \begin{align} {\mathbb{C}}[\hbar]/\langle \hbar - \lambda\rangle \otimes_{{\mathbb{C}}[\hbar]} {\mathrm{Rees}}_{\mathcal{F}} M \simeq \begin{cases} M & \text{if }\lambda \neq 0,\\ {\mathrm{gr}}_{\mathcal{F}}(M) & \text{if }\lambda=0,\\ \end{cases} \end{align} $$

where ${\mathrm {gr}}_{\mathcal {F}}(M)=\bigoplus _{j \hspace {0.1em}\mathop {\geqslant }\nolimits \hspace {0.1em} 0} {\mathcal {F}}_j M /{\mathcal {F}}_{j-1} M$ is the graded vector space associated with M and its filtration.

If A is a subalgebra of ${\mathbf H}_k$ and J is an ideal of A, then ${\mathrm {Rees}}_{\mathcal {F}}(A)$ is a subalgebra of ${\mathbb {C}}[\hbar ] \otimes A$ (called the Rees algebra of A) and ${\mathrm {Rees}}_{\mathcal {F}}(J)$ is an ideal of ${\mathrm {Rees}}_{\mathcal {F}}(A)$ . Recall [Reference Etingof and GinzburgEtGi, Theorem 1.3] that

(2-4) $$ \begin{align} {\mathrm{gr}}_{\mathcal{F}}{\mathbf H}_k \simeq {\mathbf H}_0 ={\mathbb{C}}[V \times V^*] \rtimes W \quad\text{and}\quad {\mathrm{gr}}_{\mathcal{F}} {\mathbf Z}_k \simeq {\mathbf Z}_0={\mathbb{C}}[V \times V^*]^W. \end{align} $$

2.6.4 Action of the normalizer

The group ${\mathrm {N}}_{{\mathbf G}{\mathbf L}_{\mathbb {C}}(V)}(W)$ acts on the set $\nabla $ and so on the space of parameters ${\mathbb {C}}^\nabla $ . If $\tau \in {\mathrm {N}}_{{\mathbf G}{\mathbf L}_{\mathbb {C}}(V)}(W)$ , then $\tau $ induces an isomorphism of algebras ${\mathbf H}_k \longrightarrow {\mathbf H}_{\tau (k)}$ . So, if $\tau (k)=k$ , then $\tau $ acts on the algebra ${\mathbf H}_k$ , and so on its centre ${\mathbf Z}_k$ and on the Calogero–Moser space ${\mathcal {Z}}_k$ , which preserves the ${\mathbb {C}}^\times $ -action and the Poisson bracket. We set

$$ \begin{align*}\delta(\tau)=\max_{w \in W} \dim V^{w\tau}.\end{align*} $$

Of course, $\delta (\tau )$ depends only on the coset $W\tau $ and not on $\tau $ . We say that $\tau $ is W-full if $\delta (\tau )=\dim V^\tau $ . Since W acts trivially on ${\mathcal {Z}}_k$ , the study of the action of $\tau $ on ${\mathcal {Z}}_k$ is equivalent to the study of the action of $w\tau $ . So, by replacing $\tau $ by $w\tau $ if necessary, we may assume that $\tau $ is W-full.

Example 2.2. An element $\tau \in {\mathrm {N}}_{{\mathbf G}{\mathbf L}_{\mathbb {C}}(V)}(W)$ is called W-regular (or simply regular if W is clear from the context) if $V^\tau \cap V_{\mathrm {reg}} \neq \varnothing $ . A W-regular element of ${\mathrm {N}}_{{\mathbf G}{\mathbf L}_{\mathbb {C}}(V)}(W)$ is W-full [Reference SpringerSpr].

Hypothesis and notation. From now on, and until the end of this paper, we assume that

$$ \begin{align*}W=\langle {\mathrm{Ref}}(W) \rangle,\end{align*} $$

we fix a parameter $k \in {\mathbb {C}}^\nabla $ and an element $\tau $ of finite order of ${\mathrm {N}}_{{\mathbf G}{\mathbf L}_{\mathbb {C}}(V)}(W)$ such that $\tau (k)=k$ . We also assume that $\tau $ is W-full.

If $\bigstar $ is one of the objects defined in the previous sections ( ${\mathbf H}_k$ , ${\mathcal {Z}}_k$ , $\nabla $ , ${\mathcal {A}}$ , …), we sometimes denote it by $\bigstar (W)$ or $\bigstar (V,W)$ if we need to emphasize the context.

3.1 Poisson ideals

An ideal I of R is called a Poisson ideal of R if $\{r,I\} \subset I$ for all $r \in R$ . The following facts may be found in [Reference DixmierDix, Lemma 3.3.3].

3.2 Normalization

The next result is due to Kaledin [Reference KaledinKal].

3.3 Action of a finite group

We assume in this subsection that we are given a finite group G acting on the ${\mathbb {C}}$ -algebra R in such a way that the Poisson bracket is G-equivariant (that is, $\{g(r),g(r')\}=g(\{r,r'\})$ for all $g \in G$ and r, $r' \in R$ ). Let I denote the ideal of R generated by the family $(g(r)-r)_{\substack {g \in G, r \in R}}$ . Then $R/I$ is the biggest quotient algebra of R on which G acts trivially.

Since G is finite and ${\mathbb {C}}$ has characteristic $0$ , the natural map

$$ \begin{align*}R^G \longrightarrow (R/I)^G=R/I\end{align*} $$

is surjective and its kernel is $I^G$ . Moreover, $R^G$ is a Poisson subalgebra of R (because the Poisson bracket is G-equivariant). Note that I is not in general a Poisson ideal of R, but it is easily checked that

$$ \begin{align*} I^G\ \textit{is a Poisson ideal of }R^G. \end{align*} $$

Therefore, $R/I=R^G/I^G$ can be naturally endowed with a Poisson bracket. And, by Proposition 3.1(b), $R/\!\sqrt {I}=R^G/\!\sqrt {I^G}$ also inherits a Poisson bracket.

3.4 Symplectic leaves

Assume in this subsection that $R={\mathbb {C}}[{\mathcal {X}}]$ is the coordinate ring of an affine variety ${\mathcal {X}}$ . Brown and Gordon [Reference Brown and GordonBrGo] defined a stratification of ${\mathcal {X}}$ by symplectic leaves, which are in general not algebraic subvarieties of ${\mathcal {X}}$ . We denote by ${\mathcal {S\!y\!m\!p}}({\mathcal {X}})$ the set of symplectic leaves of ${\mathcal {X}}$ .

When ${\mathcal {X}}$ has finitely many symplectic leaves, then the symplectic leaves are algebraic [Reference Brown and GordonBrGo, Proposition 3.7] and the stratification of ${\mathcal {X}}$ into symplectic leaves is given as follows. Let $({\mathcal {S}}_j)_{j \hspace {0.1em}\mathop {\geqslant }\nolimits \hspace {0.1em} 0}$ be the sequence of closed subvarieties of ${\mathcal {X}}$ defined by

$$ \begin{align*} \begin{cases} {\mathcal{S}}_0={\mathcal{X}},\\ \text{if }j \hspace{0.1em}\mathop{\geqslant}\nolimits\hspace{0.1em} 0,\text{ then }{\mathcal{S}}_{j+1}\text{ is the reduced singular locus of }{\mathcal{S}}_j. \end{cases} \end{align*} $$

Then the symplectic leaves of ${\mathcal {X}}$ are the irreducible components of the locally closed subvarieties $({\mathcal {S}}_j \setminus {\mathcal {S}}_{j+1})_{j \hspace {0.1em}\mathop {\geqslant }\nolimits \hspace {0.1em} 0}$ . Let ${\mathrm {PSpec}}({\mathbb {C}}[{\mathcal {X}}])$ denote the set of prime ideals that are Poisson. If ${\mathcal {S}}$ is a symplectic leaf of ${\mathcal {X}}$ , we denote by ${\mathfrak p}_{\mathcal {S}}$ the defining ideal of $\overline {{\mathcal {S}}}$ in ${\mathbb {C}}[{\mathcal {X}}]$ : it belongs to ${\mathrm {PSpec}}({\mathbb {C}}[{\mathcal {X}}])$ . If ${\mathcal {X}}$ has finitely many symplectic leaves, then the map

(3-1) $$ \begin{align} \begin{array}{ccc} {\mathcal{S\!y\!m\!p}}({\mathcal{X}}) \kern-2pt\!\!\!\!& \longrightarrow & \!\!\!\!{{\mathrm{PSpec}}({\mathbb{C}}[{\mathcal{X}}])}\\{{\mathcal{S}}} \kern-2pt\!\!\!\!& \longmapsto & \!\!\!\!{{\mathfrak p}_{\mathcal{S}}}\end{array} \end{align} $$

is bijective [Reference Brown and GordonBrGo, Lemma 3.4]. The inverse is given as follows: if ${\mathfrak p} \in {\mathrm {PSpec}}({\mathbb {C}}[{\mathcal {X}}])$ corresponds to ${\mathcal {S}}$ through this bijection, then ${\mathcal {S}}$ is the smooth locus of the closed irreducible subvariety of ${\mathcal {X}}$ defined by ${\mathfrak p}$ .

Now, let G be a finite group acting on ${\mathcal {X}}$ and preserving the Poisson bracket. Then ${\mathcal {X}}/G$ is an affine Poisson variety (because ${\mathbb {C}}[{\mathcal {X}}/G]={\mathbb {C}}[{\mathcal {X}}]^G$ is a Poisson subalgebra of ${\mathbb {C}}[{\mathcal {X}}]$ ; see Remark 3.3). If H is a subgroup of G, we denote by ${\mathcal {X}}(H)$ the set of elements $x \in {\mathcal {X}}$ whose stabilizer is exactly H. Then ${\mathcal {X}}(H)$ is a locally closed subvariety of ${\mathcal {X}}$ (it is open in ${\mathcal {X}}^H$ ). The subgroup H is called parabolic if ${\mathcal {X}}(H) \neq \varnothing $ . Let ${\mathrm {Parab}}(G)$ denote the set of parabolic subgroups of G.

If ${\mathfrak H}$ is a conjugacy class of parabolic subgroups of G, we denote by $({\mathcal {X}}/G)({\mathfrak H})$ the image of ${\mathcal {X}}(H)$ in ${\mathcal {X}}/G$ for some (or any) $H \in {\mathfrak H}$ . Then the group ${\overline {\mathrm {N}}}_G(H)/H$ acts freely on ${\mathcal {X}}(H)$ and the natural map ${\mathcal {X}}(H) \rightarrow ({\mathcal {X}}/G)({\mathfrak H})$ induces an isomorphism

(3-2) $$ \begin{align} {\mathcal{X}}(H)/G(H) \stackrel{\sim}{\longrightarrow} ({\mathcal{X}}/G)({\mathfrak H}). \end{align} $$

Indeed, if $g \in G$ and x, $x' \in {\mathcal {X}}(H)$ are such that $g\cdot x = x'$ , then $H=G_{x'}={^{g}G_{x}}={^{g}H}$ and so $g \in {\mathrm {N}}_G(H)$ . The next result generalizes [Reference Brown and GordonBrGo, Proposition 7.4] slightly.

Proposition 3.6. Assume that ${\mathcal {X}}$ is smooth and symplectic. Then the symplectic leaves of ${\mathcal {X}}/G$ are the irreducible components of the locally closed subvarieties $({\mathcal {X}}/G)({\mathfrak H})$ where ${\mathfrak H}$ runs over ${\mathrm {Parab}}(G)/G$ .

In particular, if all the subvarieties $({\mathcal {X}}/G)({\mathfrak H})$ are irreducible, then

$$ \begin{align*}{\mathcal{X}}/G = \dot{\bigcup_{{\mathfrak H} \in {\mathrm{Parab}}(G)/G}} ({\mathcal{X}}/G)({\mathfrak H})\end{align*} $$

is the stratification of ${\mathcal {X}}/G$ into symplectic leaves.

Proof. Let ${\mathcal {S}}$ denote a symplectic leaf of ${\mathcal {X}}$ and let $H=G_{\mathcal {S}}^{\mathrm {set}}$ . As symplectic leaves form a partition of ${\mathcal {X}}$ , and since $g({\mathcal {S}})$ is a symplectic leaf of ${\mathcal {X}}$ for any $g \in G$ , we get that

$$ \begin{align*}g({\mathcal{S}}) \cap {\mathcal{S}} = \varnothing\end{align*} $$

for all $g \not \in H$ . So, the image ${\mathcal {S}}$ in ${\mathcal {X}}/G$ is isomorphic to ${\mathcal {S}}/H$ and is a locally closed Poisson subvariety of ${\mathcal {X}}/G$ . But, by Proposition 3.6, ${\mathcal {S}}/H$ has finitely many symplectic leaves.

As ${\mathcal {X}}$ has finitely many symplectic leaves, this shows that ${\mathcal {X}}/G$ also has finitely many symplectic leaves. Now, ${\mathcal {X}}^G$ is a closed Poisson subvariety of ${\mathcal {X}}/G$ , so it also admits finitely many symplectic leaves by Lemma 3.5.

As a consequence of the above proof, we get the following corollary.

4.1 Reflection groups

Recall that $\tau \in {\mathrm {N}}_{{\mathbf G}{\mathbf L}_{\mathbb {C}}(V)}(W)$ is assumed to be W-full. This implies that [Reference SpringerSpr]

$$ \begin{align*} \delta(\tau)=\dim (V/W)^\tau. \end{align*} $$

Therefore, since $(V/W)^\tau $ is irreducible (it is isomorphic to an affine space [Reference SpringerSpr]), we get that

(4-1) $$ \begin{align} \text{the natural map }V^\tau \rightarrow (V/W)^\tau\text{ is onto.} \end{align} $$

To simplify notation, we set $W_\tau =W_{V^\tau }^{\mathrm {set}}/W_{V^\tau }^{\mathrm {pt}}$ . Note that $W^\tau \subset W_{V^\tau }^{\mathrm {set}}$ . Moreover, $W_\tau $ acts faithfully on $V^\tau $ , so

(4-2) $$ \begin{align} \tau\text{ acts trivially on }W_\tau. \end{align} $$

Lehrer–Springer theory [Reference Lehrer and SpringerLeSp, Theorem 2.5 and Corollary 2.7] gives the following result.

Similarly, the natural map $i_\tau ^{\scriptscriptstyle {\vee }} : V^{*\tau }/W_\tau \rightarrow (V^*/W)^\tau $ is an isomorphism of varieties.

4.2 ${\boldsymbol {\tau }}$ -split parabolic subgroups

A parabolic subgroup P of W is called $\tau $ -split if it is the stabilizer of some point of $V^\tau $ (that is if $P=W_{V^P \cap V^\tau }^{\mathrm {pt}}$ or, in other words, if ${\mathcal {V}}(P) \cap V^\tau \neq \varnothing $ ). This is equivalent to saying that P is the stabilizer of some point of $V^{*\tau }$ . Note the following easy fact.

$$ \begin{align*} \textit{The intersection of }\tau\textit{-split parabolic subgroups is }\tau\textit{-split.} \end{align*} $$

In this case, P is normalized by $\tau $ and $\tau $ is P-full, and we define the $\tau $ -rank of P to be the number $\dim (V^P)^\tau $ . We denote by ${\mathrm {Parab}}_\tau (W)$ the set of $\tau $ -split parabolic subgroups of W. If $P \in {\mathrm {Parab}}_\tau (W)$ , then $W_{V^\tau }^{\mathrm {pt}} \subset P$ and $P_\tau =(P \cap W_{V^\tau }^{\mathrm {set}})/W_{V^\tau }^{\mathrm {pt}}$ is a parabolic subgroup of $W_\tau $ . This shows that the map

$$ \begin{align*}\begin{array}{ccc}{{\mathrm{Parab}}_\tau(W)} \kern-1pt\!\!\!\! & \longrightarrow &\!\!\!\! {{\mathrm{Parab}}(W_\tau)}\\ {P} \kern-1pt\!\!\!\!& \longmapsto & \!\!\!\! {P_\tau}\end{array}\end{align*} $$

is well defined.

Lemma 4.3. The map

$$ \begin{align*}\begin{array}{ccc}{{\mathrm{Parab}}_\tau(W)} \kern-1pt\!\!\!\! & \longrightarrow & \!\!\!\! {{\mathrm{Parab}}(W_\tau)}\\{P} \kern-1pt\!\!\!\! & \longmapsto & \!\!\!\! {P_\tau}\end{array}\end{align*} $$

is bijective.

Proof. First, if Q is a parabolic subgroup of $W_\tau $ , then there exists $v \in V^\tau $ such that $Q=(W_\tau )_v$ and so, if we set $P=W_v$ , then P is $\tau $ -split and $P_\tau =Q$ . This shows that the map is surjective.

Now, if P is a $\tau $ -split parabolic subgroup of W, then

(4-3) $$ \begin{align} (V^P)^\tau = (V^\tau)^{P_\tau}. \end{align} $$

Proof of (4-3)

Since P is $\tau $ -split, there exists $v \in (V^P)^\tau $ such that $P=W_v$ (and so $P_\tau =(W_\tau )_v$ ). Therefore, by Theorem 4.1(a),

$$ \begin{align*}(V^\tau)^{P_\tau} = \bigcap_{\substack{H \in {\mathcal{A}}(V^\tau,W_\tau) \\ v \in H}} H,\end{align*} $$

and so, by Theorem 4.1(c),

$$ \begin{align*}(V^\tau)^{P_\tau} = V^\tau \cap \bigg(\bigcap_{\substack{H \in {\mathcal{A}}(V,W) \\ v \in H}} H \bigg)=V^\tau \cap V^P,\end{align*} $$

as expected.

Since $P=W_{(V^P)^\tau }^{\mathrm {pt}}$ , the group $P_\tau $ determines P. This means that the map of the lemma is injective.□

If ${\mathfrak Q} \in {\mathrm {Parab}}(W_\tau )/W_\tau $ and $Q \in {\mathfrak Q}$ , we denote by ${\mathcal {V}}_\tau (Q)$ and ${\mathcal {U}}_\tau ({\mathfrak Q})$ the analogues of ${\mathcal {V}}(P)$ and ${\mathcal {U}}({\mathfrak P})$ for ${\mathfrak P} \in {\mathrm {Parab}}(W)/W$ and $P \in {\mathfrak P}$ . We similarly also define ${\mathcal {V}}_\tau ^*(Q)$ and ${\mathcal {U}}_\tau ^*({\mathfrak Q})$ . The same argument as in the above proof (using Theorem 4.1(c)) shows that if P is $\tau $ -split, then

(4-4) $$ \begin{align} {\mathcal{V}}_\tau(P_\tau)={\mathcal{V}}(P)^\tau. \end{align} $$

4.3 Normalizers

Fix a $\tau $ -split parabolic subgroup P of W. If $w \in W_\tau $ , then ${^{w}P}$ does not depend on the representative of w in $W_{V^\tau }^{\mathrm {set}}$ , because $W_{V^\tau }^{\mathrm {pt}} \subset P$ by definition. So we can define the normalizer ${\mathrm {N}}_{W_\tau }(P)$ of P in $W_\tau $ and, by the bijectivity proved in Lemma 4.3, it coincides with the normalizer ${\mathrm {N}}_{W_\tau }(P_\tau )$ . The kernel of the well-defined composition

$$ \begin{align*}{\mathrm{N}}_{W_\tau}(P_\tau)={\mathrm{N}}_{W_\tau}(P) \longrightarrow {\overline{\mathrm{N}}}_W(P)/P\end{align*} $$

is equal to $P_\tau $ , so we get a natural injective map

(4-5)

Now, $\tau $ acts on ${\overline {\mathrm {N}}}_W(P)$ . The next result describes the image of the above injective map.

Thanks to Lemma 4.4, we identify ${\overline {\mathrm {N}}}_{W_\tau }(P_\tau )$ with ${\overline {\mathrm {N}}}_W(P)^\tau $ . Note that ${\overline {\mathrm {N}}}_{W_\tau }(P_\tau )={\overline {\mathrm {N}}}_W(P)^\tau $ is the stabilizer of the set ${\mathcal {V}}(P)^\tau $ in ${\overline {\mathrm {N}}}_W(P)$ .

4.4 Orbits of ${\boldsymbol {\tau }}$ -split parabolic subgroups

We denote by ${\mathrm {Parab}}(W)^\tau _{\mathrm {spl}}$ the set of $\tau $ -split parabolic subgroups of W and by $({\mathrm {Parab}}(W)/W)^\tau _{\mathrm {spl}}$ the set of W-orbits of parabolic subgroups of W containing a $\tau $ -split one. The group $W_{V^\tau }^{\mathrm {set}}$ acts on ${\mathrm {Parab}}(W)^\tau _{\mathrm {spl}}$ by conjugacy and, since any $\tau $ -split parabolic subgroup of W contains $W_{V^\tau }^{\mathrm {pt}}$ , this action factorizes through an action of $W_\tau $ . If ${\mathfrak P} \in ({\mathrm {Parab}}(W)/W)^\tau _{\mathrm {spl}}$ , we set ${\mathfrak P}^\tau _{\mathrm {spl}} = {\mathfrak P} \cap {\mathrm {Parab}}(W)_{\mathrm {spl}}^\tau $ . Now, let ${\mathcal {E}}_P$ (respectively, ${\tilde {\mathcal {E}}}_P$ ) denote the set of elements $w \in {\mathrm {N}}_W(P)$ (respectively, ${\overline {\mathrm {N}}}_W(P)$ ) such that ${\mathcal {V}}(P)^{w\tau } \neq \varnothing $ . Then ${\overline {\mathrm {N}}}_W(P)$ acts by conjugacy on the set ${\tilde {\mathcal {E}}}_P\tau $ . If $w \in {\mathcal {E}}_P$ , we denote by $[w\tau ]$ the ${\overline {\mathrm {N}}}_W(P)$ -orbit of the image of $w\tau $ in ${\overline {\mathrm {N}}}_W(P)\tau $ .

Proof. (a) Assume that ${^{x}P}$ is $\tau $ -split. In other words, there exists $v \in V^\tau $ such that ${^{x}P}=W_v$ . Now let $w=x^{-1}\tau (x)$ . Then $x^{-1}(v) \in {\mathcal {V}}(P)^{w\tau }$ and so $w \in {\mathcal {E}}_P$ .

Conversely, assume that $w=x^{-1}\tau (x) \in {\mathcal {E}}_P$ . Then there exists $v \in {\mathcal {V}}(P)$ such that $v \in V^{w\tau }$ . Therefore, $P=W_v$ and so ${^{x}P}=W_{x(v)}$ . But $\tau (x(v))=xx^{-1}\tau (x)\tau (v)=xw\tau (v)=x(v)$ , so $x(v) \in V^\tau $ . This implies that ${^{x}P}$ is $\tau $ -split by definition.

(b) Let us first show that the map (let us denote it by $\phi $ ) is well defined. For this purpose, let x and y be two elements of W such that ${^{x}P}={^{y}P}$ is $\tau $ -split. Then there exists $u \in {\mathrm {N}}_W(P)$ such that $y=xu$ . So $y^{-1}\tau y = u^{-1} x^{-1} \tau x u$ and so $[\,y^{-1}\tau y]=[x^{-1}\tau x]$ , as expected.

Let us now prove that $\phi $ is constant on $W_\tau $ -orbits. For this purpose, let $w \in W_\tau $ and $x \in W$ be such that $\xi ^{-1}\tau (x) \in {\mathcal {E}}_P$ . Then $(xw)^{-1}\tau (wx)=x^{-1} w^{-1}\tau (w) x x^{-1}\tau (x)$ . But $w^{-1}\tau (w) \in W_{V^\tau }^{\mathrm {pt}} \subset P$ by (4-2), so the images of $x^{-1}\tau (x)$ and $(wx)^{-1}\tau (wx)$ in ${\tilde {\mathcal {E}}}_P$ coincide. Therefore, $\phi $ factorizes through a map

$$ \begin{align*}{\tilde{\phi}} : {\mathfrak P}_{\mathrm{spl}}^\tau/W_\tau \longrightarrow {\mathcal{E}}_P\tau/{\overline{\mathrm{N}}}_W(P).\end{align*} $$

Let us prove that ${\tilde {\phi }}$ is injective. So let x and y be two elements of W such that ${^{x}P}$ and ${^{y}P}$ are $\tau $ -split and $[x^{-1}\tau x]=[\,y^{-1}\tau y]$ . Then there exists $u \in {\mathrm {N}}_W(P)$ and $p \in P$ such that $y^{-1}\tau y = u^{-1}x^{-1}\tau x u p$ . In particular, ${\mathcal {V}}(P)^{y^{-1}\tau y}={\mathcal {V}}(P)^{u^{-1}x^{-1}\tau x u}$ . Since ${^{x}P}={^{xu}P}$ , we may (and do) assume that $u=1$ . As ${^{x}P}$ is $\tau $ -split, the set ${\mathcal {V}}({^{x}P})^\tau $ is nonempty, so we may pick an element $v \in {\mathcal {V}}({^{x}P})^\tau $ . Then

$$ \begin{align*}\tau yx^{-1}(v)=yy^{-1}\tau y x^{-1}(v)=yx^{-1}\tau x p x^{-1}(v).\end{align*} $$

But $x^{-1}(v) \in {\mathcal {V}}(P)$ , so $px^{-1}(v)=x^{-1}(v)$ . Consequently, $\tau yx^{-1}(v)= yx^{-1} \tau (v)=yx^{-1}(v)$ . In other words, $yx^{-1}(v) \in {\mathcal {V}}({^{y}P})^\tau \subset V^\tau $ . By Lehrer–Springer Theorem 4.1(b), there exists $a \in W_\tau $ such that $yx^{-1}(v)=a(v)$ . Then

$$ \begin{align*}{^{y}P}={^{yx^{-1}}}{(^{x}P)}={^{yx^{-1}}} W_v =W_{yx^{-1}(v)}=W_{a(v)}={^{a}W_{v}}={{^{a}}(^{x}{P})},\end{align*} $$

which shows that ${^{y}P}$ and ${^{x}P}$ are $W_\tau $ -conjugate.

Let us now prove that ${\tilde {\phi }}$ is surjective. So let $w \in {\mathcal {E}}_P$ . Then there exists $v \in {\mathcal {V}}(P)^{w\tau }$ . So $W \cdot v \in (V/W)^\tau $ . By Lehrer–Springer Theorem 4.1, there exists $x \in W$ such that $x(v) \in V^\tau $ . Therefore, ${^{x}P}=W_{x(v)}$ is $\tau $ -split and $v \in V^{x^{-1}\tau x}$ . So, if we set $p=w^{-1}x^{-1}\tau (x)$ , then $p(v)=v$ ; so $p \in P$ and $\phi ({^{x}P})=[x^{-1}\tau (x)]=[wp]=[w]$ , as desired.

4.5 Stratification of ${\boldsymbol {(\mathit {V/W})^\tau }}$

Applying Section 2.3 to the pair $(V^\tau ,W_\tau )$ , the variety $V^\tau /W_\tau $ admits a stratification $({\mathcal {U}}_\tau ({\mathfrak Q}))_{{\mathfrak Q} \in {\mathrm {Parab}}(W_\tau )/W_\tau }$ while the variety $(V/W)^\tau $ admits a stratification $({\mathcal {U}}({\mathfrak P})^\tau )_{{\mathfrak P} \in {\mathrm {Parab}}(W)/W}$ . Both varieties are isomorphic and so both stratifications can be compared: through this isomorphism, the first is a refinement of the second, as shown in Corollary 4.7 below by using Proposition 4.5.

After we eliminate the empty pieces, Proposition 4.6 shows that $(V/W)^\tau $ admits a stratification $({\mathcal {U}}({\mathfrak P})^\tau )_{{\mathfrak P} \in ({\mathrm {Parab}}(W)/W)_{\mathrm {spl}}^\tau }$ . Let us decompose the pieces of this stratification into irreducible components. For this, fix a $\tau $ -split parabolic subgroup P and let ${\mathfrak P}$ denote its conjugacy class. Then ${\mathcal {U}}({\mathfrak P})^\tau $ is smooth since ${\mathcal {U}}({\mathfrak P})$ is smooth and $\tau $ has finite order. Now we have

$$ \begin{align*}{\mathcal{U}}({\mathfrak P})^\tau = ({\mathcal{V}}(P)/{\overline{\mathrm{N}}}_W(P))^\tau= \bigg(\bigcup_{w \in {\tilde{\mathcal{E}}}_P} {\mathcal{V}}(P)^{w\tau}\bigg)/{\overline{\mathrm{N}}}_W(P).\end{align*} $$

By definition of ${\mathcal {V}}(P)$ , ${\mathcal {V}}(P)^{w\tau } \cap {\mathcal {V}}(P)^{w'\tau }=\varnothing $ if $w \neq w'$ . If E is a subset of ${\tilde {\mathcal {E}}}_P \tau $ , we denote by ${\mathcal {V}}(P)^E$ the (disjoint) union of the ${\mathcal {V}}(P)^g$ for $g \in E$ . Then

(4-6) $$ \begin{align} {\mathcal{U}}({\mathfrak P})^\tau = \bigcup_{E \in {\tilde{\mathcal{E}}}_P\tau/{\overline{\mathrm{N}}}_W(P)} {\mathcal{V}}(P)^E/{\overline{\mathrm{N}}}_W(P). \end{align} $$

Then ${\mathcal {V}}(P)^E/{\overline {\mathrm {N}}}_W(P)$ is the image of some ${\mathcal {V}}(P)^g$ for some $g \in E$ and so ${\mathcal {V}}(P)^E/{\overline {\mathrm {N}}}_W(P)$ is closed (in ${\mathcal {U}}({\mathfrak P})$ ) and irreducible. So the decomposition (4-6) is the decomposition of ${\mathcal {U}}({\mathfrak P})^\tau $ into irreducible (that is, connected because disjoint) components.

So the stratification $({\mathcal {U}}({\mathfrak P})^\tau )_{{\mathfrak P} \in ({\mathrm {Parab}}(W)/W)_{\mathrm {spl}}^\tau }$ of $(V/W)^\tau $ together with the decomposition (4-6) provides a finer stratification of $(V/W)^\tau $ , indexed by the $W_\tau $ -orbits of $\tau $ -split parabolic subgroups (by using the bijection of Proposition 4.5(b)). On the other hand, $V^\tau /W_\tau $ admits a stratification $({\mathcal {U}}_\tau ({\mathfrak Q}))_{{\mathfrak Q} \in {\mathrm {Parab}}(W_\tau )/W_\tau }$ . Both stratifications coincide through the isomorphism $i_\tau $ , as shown by the next result.

Corollary 4.7. Let ${\mathfrak P} \in ({\mathrm {Parab}}(W)/W)_{\mathrm {spl}}^\tau $ , let $P \in {\mathfrak P}$ and let $E \in {\tilde {\mathcal {E}}}_P\tau /{\overline {\mathrm {N}}}_W(P)$ . Let ${\mathfrak P}_E$ denote the $W_\tau $ -orbit of $\tau $ -split parabolic subgroups of W associated with E through the bijection of Proposition 4.5(b). Let ${\mathfrak Q}_E$ denote the $W_\tau $ -orbit of parabolic subgroups of $W_\tau $ of the form $Q_\tau $ for $Q \in {\mathfrak P}_E$ (see Lemma 4.3). Then

$$ \begin{align*}i_\tau({\mathcal{U}}_\tau({\mathfrak Q}_E))={\mathcal{V}}(P)^E/{\overline{\mathrm{N}}}_W(P).\end{align*} $$

5.1 Symplectic leaves

Let $I_k$ denote the ideal of ${\mathbf Z}_k$ generated by $(\tau (z)-z)_{z \in {\mathbf Z}_k}$ . It is $\tau $ -stable. Recall from Remark 3.3 that ${\mathbb {C}}[{\mathcal {Z}}_k^\tau ] = {\mathbf Z}_k /\!\sqrt {I_k}={\mathbf Z}_k^\tau /\!\sqrt {I_k^\tau }$ , and that ${\mathbf Z}_k/I_k$ inherits a Poisson bracket which makes ${\mathcal {Z}}_k^\tau $ an affine Poisson variety. Therefore, ${\mathcal {Z}}_k^\tau $ admits a stratification into symplectic leaves [Reference Brown and GordonBrGo, Section S.5]. We denote by ${\mathcal {S\!y\!m\!p}}({\mathcal {Z}}_k^\tau )$ the set of symplectic leaves of ${\mathcal {Z}}_k^\tau $ .

Since ${\mathcal {Z}}_k$ has finitely many symplectic leaves [Reference Brown and GordonBrGo, Proposition 7.4], it follows from Corollary 3.7 that ${\mathcal {Z}}_k^\tau $ has finitely many symplectic leaves too. They are obtained as in Section 3.4.

If ${\mathcal {S}}$ is a symplectic leaf of ${\mathcal {Z}}_k^\tau $ , we denote by ${\mathfrak p}_{\mathcal {S}}$ the defining ideal of $\overline {{\mathcal {S}}}$ in ${\mathbf Z}_k/I_k$ : it belongs to ${\mathrm {PSpec}}({\mathbf Z}_k/I_k)$ . Since ${\mathcal {Z}}_k^\tau $ has finitely many symplectic leaves, the map

(5-1) $$ \begin{align} \begin{array}{ccc}{{\mathcal{S\!y\!m\!p}}({\mathcal{Z}}_k^\tau)}\kern-7pt & \longrightarrow &\kern-8pt{{\mathrm{PSpec}}({\mathbf Z}_k/I_k)}\\{{\mathcal{S}}}\kern-7pt& \,\longmapsto &\kern-8pt{{\mathfrak p}_{\mathcal{S}}}\end{array} \end{align} $$

is bijective (see (3-1)).

5.2 ${\boldsymbol {\tau }}$ -cuspidality

We define a $\tau $ -cuspidal symplectic leaf to be a zero-dimensional symplectic leaf of ${\mathcal {Z}}_k^\tau $ . This definition coincides with the notion of cuspidal leaf of ${\mathcal {Z}}_k$ introduced by Bellamy [Reference BellamyBel1, Section 5] in the case where $\tau =1$ . We therefore also call it a $\tau $ -cuspidal point. Through the bijection (5-1), the set of $\tau $ -cuspidal points is naturally in bijection with the set ${\mathrm {PMax}}({\mathbf Z}_k/\!\sqrt {I_k})$ of maximal ideals of the algebra ${\mathbb {C}}[{\mathcal {Z}}_k^\tau ]={\mathbf Z}_k/\!\sqrt {I_k}$ that are also Poisson ideals (note that ${\mathrm {PMax}}({\mathbf Z}_k/I_k)= {\mathrm {PMax}}({\mathbf Z}_k/\!\sqrt {I_k})\subset {\mathrm {PSpec}}({\mathbf Z}_k/\!\sqrt {I_k})$ ).

We denote by $\operatorname {{{\mathrm {Cus}}}}_k^\tau (V,W)$ the set of pairs $(P,p)$ where P is a $\tau $ -split parabolic subgroup of W and p is a $\tau $ -cuspidal point of ${\mathcal {Z}}_{k_P}(V_P,P)^\tau $ , where $k_P$ denotes the restriction of k to the parabolic subgroup P. The group $W_\tau $ acts on $\operatorname {{{\mathrm {Cus}}}}_k^\tau (V,W)$ and we denote by $\operatorname {{{\mathrm {Cus}}}}_k^\tau (V,W)/W_\tau $ the set of its orbits in $\operatorname {{{\mathrm {Cus}}}}_k^\tau (V,W)$ . If $(P,p) \in \operatorname {{{\mathrm {Cus}}}}_k^\tau (V,W)$ , we denote by $[P,p]$ its $W_\tau $ -orbit.

5.3 Main result

With the above notation, Theorem A can be restated (and made more precise) as follows.

Theorem A. There is a natural bijection (explicitly constructed in Section 9)

$$ \begin{align*}\begin{array}{ccc}{\operatorname{{{\mathrm{Cus}}}}_k^\tau(V,W)/W_\tau}\kern-7pt& \longrightarrow &\kern-7pt{{\mathcal{S\!y\!m\!p}}({\mathcal{Z}}_k^\tau)}\\{[P,p]}\kern-7pt& \longmapsto &\kern-7pt{\mathcal{S}}_{P,p}.\end{array}\end{align*} $$

It satisfies that $\Upsilon _k({\overline {{\mathcal {S}}}}_{P,p})$ is the image of $(V^P)^\tau \times (V^{*P})^\tau $ in $V/W \times V^*/W$ . In particular,

$$ \begin{align*}\dim {\mathcal{S}}_{P,p} = 2 \dim (V^P)^\tau.\end{align*} $$

We prove Theorem A in the next sections. First, in Section 6, we recall the proof, essentially due to Brown and Gordon [Reference Brown and GordonBrGo, Proposition 7.4], of Theorem A whenever $k=0$ and $\tau =1$ . In Section 7 we use Lehrer–Springer Theorem 4.1 to prove Theorem A whenever $k=0$ . In Section 8 we use a deformation argument to attach to each symplectic leaf a $W_\tau $ -orbit of $\tau $ -split parabolic subgroups: in some sense, this is half of the construction of the above bijection. The second half is constructed in Section 9, where the proof of Theorem A is completed.

Let us also restate Conjecture B.

Conjecture B. Let $(P,p) \in \operatorname {{{\mathrm {Cus}}}}_k^\tau (V,W)$ . Then there exist $l \in \nabla ((V^P)^\tau ,{\overline {\mathrm {N}}}_{W_\tau }(P_\tau ))$ and a ${\mathbb {C}}^\times $ -equivariant isomorphism of Poisson varieties

$$ \begin{align*}\overline{{\mathcal{S}}}_{P,p}^{\mathrm{nor}} \simeq {\mathcal{Z}}_l((V^P)^\tau,{\overline{\mathrm{N}}}_{W_\tau}(P_\tau)).\end{align*} $$

8.1 Definition

Let ${\mathcal {S}}$ be a symplectic leaf of ${\mathcal {Z}}_k^\tau $ . We denote by ${\mathfrak p}_{\mathcal {S}}$ the prime ideal of ${\mathbf Z}_k$ defining ${\overline {{\mathcal {S}}}}$ ; then ${\mathfrak p}_{\mathcal {S}}^\tau \in {\mathrm {PSpec}}({\mathbf Z}_k^\tau )$ . Now, the isomorphism ${\mathrm {gr}}_{\mathcal {F}} {\mathbf Z}_k \simeq {\mathbf Z}_0$ is $\tau $ -equivariant and $({\mathrm {gr}}_{\mathcal {F}} {\mathbf Z}_k)^\tau ={\mathrm {gr}}_{\mathcal {F}}({\mathbf Z}_k^\tau )$ . So ${\mathrm {gr}}_{\mathcal {F}}({\mathfrak p}_{\mathcal {S}}^\tau )$ is an ideal of ${\mathbf Z}_0^\tau $ . The next important result follows mainly from [Reference MartinoMar1, Theorem 2.8].

Lemma 8.1 shows that $\!\!\sqrt {{\mathrm {gr}}_{\mathcal {F}}({\mathfrak p}_{\mathcal {S}}^\tau )}$ defines a symplectic leaf ${\mathcal {S}}_0$ of ${\mathcal {Z}}_0^\tau $ ; so, by Corollary 7.5, there exists a unique $W_\tau $ -orbit ${\mathfrak Q}_{\mathcal {S}}$ of parabolic subgroups of $W_\tau $ such that $\!\sqrt {{\mathrm {gr}}_{\mathcal {F}}({\mathfrak p}_{\mathcal {S}}^\tau )}$ is the defining ideal of $i\hskip -0.3mm i_\tau ({\mathcal {U}}\!{\mathcal {U}}_\tau ^*({\mathfrak Q}_{\mathcal {S}}))$ . Through the bijection of Lemma 4.3, there exists a unique $W_\tau $ -orbit ${\mathfrak P}_{\mathcal {S}}$ of $\tau $ -split parabolic subgroups of W such that ${{\mathfrak Q}_{\mathcal {S}}=\{P_\tau ~|~P \in {\mathfrak P}_{\mathcal {S}}\}}$ .

If P is a $\tau $ -split parabolic subgroup of W associated with ${\mathcal {S}}$ , then

(8-1) $$ \begin{align} \dim {\mathcal{S}} = 2 \dim(V^P)^\tau. \end{align} $$

Indeed, $\dim {\mathcal {S}} = \dim {\mathcal {S}}_0$ .

8.2 Geometric construction

Let

$$ \begin{align*}\pi : V/W \times V^*/W \rightarrow V/W \quad \text{and} \quad \pi ^{\scriptscriptstyle {\vee }} : V/W \times V^*/W \longrightarrow V^*/W\end{align*} $$

denote the first and second projection, respectively. The next proposition gives another characterization of the $W_\tau $ -orbit of $\tau $ -split parabolic subgroups associated with a symplectic leaf:

Proof. Let ${\mathbf H}_k^\#$ denote the ${\mathbb {C}}[\hbar ]$ -algebra that is obtained as the quotient of the algebra ${\mathbb {C}}[\hbar ] \otimes ({\mathrm {T}}(V \oplus V^*) \rtimes W)$ by the relations

(8-2) $$ \begin{align} \begin{cases} [x,x']=[\,y,y']=0,\\ [\,y,x]=\hbar^2 \displaystyle{\sum_{H\in\mathcal{A}} \sum_{j=0}^{e_H-1} e_H(k_{H,i}-k_{H,i+1}) \frac{\langle y,\alpha_H \rangle \cdot \langle \alpha_H^{\scriptscriptstyle{\vee}},x\rangle}{\langle \alpha_H^{\scriptscriptstyle{\vee}},\alpha_H\rangle} \varepsilon_{H,i}}, \end{cases} \end{align} $$

for all y, $y' \in V$ and x, $x' \in V^*$ . It follows from the comparison of the relations (2-1) and (8-2) that there is a well-defined morphism of ${\mathbb {C}}[\hbar ]$ -algebras $\theta : {\mathbf H}_k^\# \longrightarrow {\mathrm {Rees}}_{\mathcal {F}} {\mathbf H}_k$ such that

$$ \begin{align*}\theta(y)=\hbar y,\quad \theta(x)=\hbar x\quad \text{and}\quad \theta(w)=w\end{align*} $$

for all $y \in V$ , $x \in V^*$ and $w \in W$ . In fact,

(8-3) $$ \begin{align} \theta\text{ is an isomorphism of algebras.} \end{align} $$

Indeed, surjectivity is immediate while injectivity follows from the PBW decomposition (2-2), which also holds for ${\mathbf H}_k^\#$ , namely, the map

$$ \begin{align*}{\mathbb{C}}[\hbar] \otimes {\mathbb{C}}[V] \otimes {\mathbb{C}} W \otimes {\mathbb{C}}[V^*] \longrightarrow {\mathbf H}_k^\#\end{align*} $$

induced by the multiplication is an isomorphism of ${\mathbb {C}}[\hbar ]$ -modules.

Let ${\mathbf Z}_k^\#$ denote the centre of ${\mathbf H}_k^\#$ . Then it follows from (8-3) that $\theta $ induces an isomorphism of algebras

$$ \begin{align*} {\mathbf Z}_k^\# \longrightarrow {\mathrm{Rees}}_{\mathcal{F}} {\mathbf Z}_k. \end{align*} $$

Again, ${\mathbf Z}_k^\#$ is a flat family of deformations of ${\mathbf Z}_0={\mathbb {C}}[V \times V^*]^W$ . We denote by ${\mathcal {Z}}_k^\#$ the affine variety such that ${\mathbb {C}}[{\mathcal {Z}}_k^\#]={\mathbf Z}_k^\#$ . The inclusion ${\mathbf P} \hookrightarrow {\mathbf Z}_k^\#$ induces a morphism $\Upsilon _k^\# : {\mathcal {Z}}_k^\# \longrightarrow {\mathcal {P}}=V/W \times V^*/W$ .

The action of $\tau $ and ${\mathbb {C}}^\times $ extends easily to ${\mathbf H}_k^\#$ , by letting them act trivially on the indeterminate $\hbar $ . However, ${\mathbf H}_k^\#$ (and so ${\mathbf Z}_k^\#$ ) inherits a further action of ${\mathbb {C}}^\times $ . Namely, there is an action of ${\mathbb {C}}^\times \times {\mathbb {C}}^\times $ given by

$$ \begin{align*} \begin{cases} (\xi,\xi') \cdot x = \xi\xi' x & \text{if }x \in V^*,\\ (\xi,\xi') \cdot y = \xi^{-1}\xi' y & \text{if }y \in V,\\ (\xi,\xi') \cdot w = w & \text{if }w \in W,\\ (\xi,\xi') \cdot \hbar = \xi'\hbar. \end{cases}\end{align*} $$

The action of the first copy of ${\mathbb {C}}^\times $ extends the one already defined in Section 2.6.1. Through the isomorphism $\theta $ , this action on ${\mathrm {Rees}}_{\mathcal {F}} {\mathbf H}_k$ is just the restriction of the action on ${\mathbb {C}}[\hbar ] \otimes {\mathbf H}_k$ given by

$$ \begin{align*}(\xi,\xi') \cdot (P(\hbar) \otimes h)=P(\xi'\hbar) \otimes (\xi \cdot h).\end{align*} $$

Then (2-4) can be retrieved thanks to (2-3) and the isomorphism $\theta $ , by specializing $\hbar $ to $0$ .

Specializing $\hbar $ to $\lambda \in {\mathbb {C}}$ gives the algebras ${\mathbf H}_{\lambda ^2k}$ and ${\mathbf Z}_{\lambda ^2 k}$ . Geometrically, the inclusion ${\mathbb {C}}[\hbar ] \hookrightarrow {\mathbf Z}_k^\#$ induces a flat morphism ${\mathcal {Z}}_k^\# \rightarrow {\mathbb {C}}$ whose fibre at $\lambda $ is the Calogero–Moser space ${\mathcal {Z}}_{\lambda ^2 k}$ .

Now view ${\mathcal {S}}$ as a subvariety of ${\mathcal {Z}}_k^\#$ and let ${\overline {{\mathcal {S}}}}_0=\overline {(1 \times {\mathbb {C}}^\times )\cdot {\mathcal {S}}} \cap {\mathcal {Z}}_0$ , endowed with its reduced structure. Then, using the isomorphism $\theta $ , it follows from the definition of the action of the second copy of ${\mathbb {C}}^\times $ on ${\mathrm {Rees}}_{\mathcal {F}}({\mathbf H}_k)$ that the defining ideal of $\overline {{\mathcal {S}}}_0$ is $\!\sqrt {{\mathrm {gr}}_{\mathcal {F}}({\mathfrak p}_{\mathcal {S}})}$ .

Since ${\mathcal {S}}$ is $({\mathbb {C}}^\times \times 1)$ -stable, we have

$$ \begin{align*}(1 \times {\mathbb{C}}^\times)\cdot {\mathcal{S}}=({\mathbb{C}}^\times \times {\mathbb{C}}^\times)\cdot {\mathcal{S}} = \Delta({\mathbb{C}}^\times) \cdot {\mathcal{S}},\end{align*} $$

where $\Delta ({\mathbb {C}}^\times )$ is the diagonal in ${\mathbb {C}}^\times \times {\mathbb {C}}^\times $ . Note also that if $\xi \in {\mathbb {C}}^\times $ and $z \in {\mathcal {Z}}_k^\#$ , then $\pi \circ \Upsilon _k^\#((\xi ,\xi ) \cdot z)=\pi \circ \Upsilon _k^\#(z)$ . Moreover, if $z \in \overline {{\mathcal {S}}}$ , then $z_0=\lim _{\xi \rightarrow 0} (\xi ,\xi ) \cdot z$ exists (because the action $\Delta ({\mathbb {C}}^\times )$ has nonnegative weights) and $z_0 \in \overline {{\mathcal {S}}}_0$ . So $(\pi \circ \Upsilon _k)(z)=(\pi \circ \Upsilon _0)(z_0)$ belongs to $\iota _\tau (\overline {{\mathcal {U}}_\tau ({\mathfrak Q}_{\mathcal {S}})})$ , as expected. This shows that

$$ \begin{align*}\pi(\Upsilon_k(\overline{{\mathcal{S}}})) \subset \iota_\tau(\overline{{\mathcal{U}}_\tau({\mathfrak Q}_{\mathcal{S}})}).\end{align*} $$

By exchanging the role of V and $V^*$ , we have

$$ \begin{align*}\pi^{\scriptscriptstyle{\vee}}(\Upsilon_k(\overline{{\mathcal{S}}})) \subset \iota_\tau^{\scriptscriptstyle{\vee}}(\overline{{\mathcal{U}}_\tau^*({\mathfrak Q}_{\mathcal{S}})}).\end{align*} $$

Therefore,

$$ \begin{align*}\Upsilon_k(\overline{{\mathcal{S}}}) \subset \iota_\tau(\overline{{\mathcal{U}}_\tau({\mathfrak Q}_{\mathcal{S}})}) \times \iota_\tau^{\scriptscriptstyle{\vee}}(\overline{{\mathcal{U}}_\tau^*({\mathfrak Q}_{\mathcal{S}})}).\end{align*} $$

Since $\Upsilon _k({\overline {{\mathcal {S}}}})$ is closed irreducible of dimension $2 \dim (V^P)^\tau $ (by (8-1) and the finiteness of the morphism $\Upsilon _k$ ), we get that

$$ \begin{align*}\Upsilon_k(\overline{{\mathcal{S}}}) = \iota_\tau(\overline{{\mathcal{U}}_\tau({\mathfrak Q}_{\mathcal{S}})}) \times \iota_\tau^{\scriptscriptstyle{\vee}}(\overline{{\mathcal{U}}_\tau^*({\mathfrak Q}_{\mathcal{S}})}).\end{align*} $$

In other words, this proves (a). Now, (b) and (c) follow from (a).

Keep the notation introduced in the above proof ( ${\mathbf H}_k^\#$ , ${\mathbf Z}_k^\#$ , ${\mathcal {Z}}_k^\#$ , …) and let us explain how this proof provides a justification for Conjecture B as well as a possible strategy for proving it. Indeed, if ${\mathcal {S}}$ is a symplectic leaf of ${\mathcal {Z}}_k$ , let ${\overline {{\mathcal {S}}}}^\#=\overline {(1 \times {\mathbb {C}}^\times )\cdot {\mathcal {S}}}$ . Then ${\overline {{\mathcal {S}}}}^\#$ comes equipped with a morphism $\varpi : {\overline {{\mathcal {S}}}}^\# \longrightarrow {\mathbb {C}}$ and we denote by $\nu : ({\overline {{\mathcal {S}}}}^\#)^{\mathrm {nor}} \longrightarrow {\mathbb {C}}$ the composition of the normalization morphism $({\overline {{\mathcal {S}}}}^\#)^{\mathrm {nor}} \longrightarrow {\overline {{\mathcal {S}}}}^\#$ with $\varpi $ . Then $\nu $ is flat [Reference HartshorneHar, Ch. III, Proposition 9.7]. Since $\varpi ^{-1}({\mathbb {C}}^\times ) \simeq {\mathbb {C}}^\times \times {\overline {{\mathcal {S}}}}$ , we have

$$ \begin{align*}\nu^{-1}({\mathbb{C}}^\times) \simeq {\mathbb{C}}^\times \times {\overline{{\mathcal{S}}}}^{\mathrm{nor}}.\end{align*} $$

Let ${\overline {{\mathcal {S}}}}_0^\bigstar $ denote the scheme-theoretic fibre of $\nu $ at $0$ . Assume that we are able to show the following two facts.

1. (1) The reduced subscheme of ${\overline {{\mathcal {S}}}}_0^\bigstar $ is the normalization of ${\overline {{\mathcal {S}}}}_0$ .

2. (2) The scheme ${\overline {{\mathcal {S}}}}_0^\bigstar $ is generically reduced.

Then a theorem of Hironaka [Reference HartshorneHar, Ch. III, Theorem 9.11] would show that $\nu $ is a flat family of schemes, all of whose scheme-theoretic fibres are reduced, irreducible and normal varieties. As ${\overline {{\mathcal {S}}}}_0^\bigstar ={\overline {{\mathcal {S}}}}_0^{\mathrm {nor}}$ is the normalization of ${\overline {{\mathcal {S}}}}_0$ by (1), this would imply that ${\overline {{\mathcal {S}}}}^{\mathrm {nor}}$ is a Poisson deformation of ${\overline {{\mathcal {S}}}}_0^{\mathrm {nor}}$ . So Conjecture B would then follow from Propositions 7.4, 7.6 and a result of Bellamy [Reference BellamyBel2, Theorem 1.4] (which follows works by Ginzburg and Kaledin [Reference Ginzburg and KaledinGiKa] and Namikawa [Reference NamikawaNam1, Reference NamikawaNam2]).

10.1 Smooth case

Assume in this subsection, and only in this subsection, that ${\mathcal {Z}}_k$ is smooth and that $\tau $ is of the form $\zeta w$ for some root of unity $\zeta $ and $w \in W$ . We denote by d the order of $\zeta $ . Then

$$ \begin{align*} {\mathcal{Z}}_k^\tau = {\mathcal{Z}}_k^{{\boldsymbol{\mu}}_d}. \end{align*} $$

Since ${\mathcal {Z}}_k$ is smooth, it is symplectic by (5-1), and so ${\mathcal {Z}}_k^\tau $ is also smooth and symplectic: its symplectic leaves are exactly its irreducible (that is, connected) components.

In [Reference Bonnafé and MaksimauBoMa], Maksimau and the author have described the irreducible components of ${\mathcal {Z}}_k^{{\boldsymbol {\mu }}_d}$ as particular Calogero–Moser spaces, and the reader can check that this description is compatible with Conjecture B; however, it is not proved that the isomorphism preserves the Poisson structure. In other words, we have the following result [Reference Bonnafé and MaksimauBoMa, Theorems 2.13 and 5.1].

10.2 Type ${\boldsymbol {\mathit {G}_4}}$

Thiel and the author [Reference Bonnafé and ThielBoTh] have developed algorithms for computing presentations of ${\mathbf Z}_k$ that have been implemented in Magma [Reference Bosma, Cannon and PlayoustMag] (more precisely, in the Champ package for Magma written by Thiel [Reference ThielThi]). This allows computations for (very) small groups.

We assume in this subsection, and only in this subsection, that W is the group $G_4$ , in the Shephard–Todd classification [Reference Shephard and ToddShTo]. Then a presentation of ${\mathbf Z}_k$ can be obtained with Magma (see, for instance, [Reference Bonnafé and MaksimauBoMa, Section 5] or [Reference Bonnafé and ThielBoTh, Theorem 5.2]), and it has been checked in [Reference Bonnafé and ThielBoTh, Theorem 4.7] that Conjecture B holds in this case.

10.3 Type ${\boldsymbol {\mathit {B}}}$

Assume in this subsection, and only in this subsection, that $W=W_n$ is a Coxeter group of type $B_n$ for some $n \hspace {0.1em}\mathop {\geqslant }\nolimits \hspace {0.1em} 2$ (that is, we may assume that $W=G(2,1,n)$ in the Shephard–Todd classification) and that $\tau =\operatorname {{{\mathrm {Id}}}}_V$ . Let $t=\operatorname {{{\mathrm {diag}}}}(-1,1,\ldots ,1) \in W_n$ and, for $1 \hspace {0.1em}\mathop {\leqslant }\nolimits \hspace {0.1em} j \hspace {0.1em}\mathop {\leqslant }\nolimits \hspace {0.1em} n-1$ , let $s_j$ denote the permutation matrix corresponding to the transposition $(\,j,j+1)$ .

There are two conjugacy classes of reflections: the class of t (which generates an elementary abelian normal subgroup of order $2^n$ ) and that of $s_1$ (which generates a normal subgroup $W_n'=G(2,2,n)$ of index $2$ isomorphic to a Coxeter group of type $D_n$ ). We set $b=c_k(t)$ and $a=c_k(s_1)$ and we denote by $I_n$ the set of $m \in {\mathbb {Z}}$ such that $|m| \hspace {0.1em}\mathop {\leqslant }\nolimits \hspace {0.1em} n-1$ . The Dynkin diagram, together with the values of the parameter function $c_k$ , is given as follows:

The case where $a=0$ is somewhat uninteresting, as then ${\mathcal {Z}}_k \simeq (C_b)^n/{\mathfrak S}_n$ , where $C_b$ is the Calogero–Moser space associated with the cyclic group of order $2$ whose equation is given by $C_b=\{(x,y,z) \in {\mathbb {C}}^3~|~z^2=xy+4b^2\}$ . So we assume throughout this subsection that $a \neq 0$ . This implies that